Let $G$ be a finite group that is neither cyclic, nor a direct product of cyclic groups. Suppose $g \in G$ is an element of order $\frac{|G|}{2}$. Then what could be said about the structure of $G$?
Dihedral groups are examples of this class of groups and I was wondering if more structural properties could be determined based on this property.
Thank you for your insights!
$G$ must be of the following form: $G=C\times (D \rtimes P_2)$, where $C$ and $D$ are cyclic of odd coprime order, $P_2$ is a $2$-group having a cyclic subgroup of index $2$, say $C_2$, and $P_2$ acts on $D$ by inversion, with kernel $C_2$.
(In other words, every element in $C_2$ centralises $D$, whereas every element in $P_2$ outside $C_2$ acts by inversion on $D$.)
Note that the possibilities for $P_2$ are well-known (see comments for example). In particular, for a given order, there are at most $6$ isomorphisms types.
Note that every $G$ of this form has the property you want, so this is a complete characterisation. The proof goes roughly as in the comments.